Alessandro Sarti !joint work with Giovanna Citti

Phenomenological Gestalten and figural completion: A neurogeometrical approach

Center of Mathematics!CNRS-EHESS, Paris!Equipe Neuromathématiques

Modal completion

Amodal completion

Amodal completion

The hypercolumnar module

The pinwheel structure

The Cortex as a fiber bundle

W.Hoffman, J.Koenderink, S.Zucker, Bressloff Cowan, J. Petitot, Citti-Sarti, R.Duits, Boscain-Gauthier

⇡ : G ! B

C = (G,⇡, B)

The Cortex as a fiber bundle on the Lie group SE(2)

C = (G,⇡, B) = (E(2),⇡, R2)

Infinitesimal transformation and the Lie algebra

The stratified Lie algebra of SE(2) and the sub-Riemannian structure

X1 = cos(✓)@x

+ sin(✓)@y

X3 = [X2, X1] = �sin(✓)@x

+ cos(✓)@y

X2 = @✓

are left invariant for E(2)X1, X2, X3

The Hormander condition holds

Sarti , Citti 2003 Citti, Sarti 2006

The integral curves of the algebra

≈

Horizontal Connectivity

The neurogeometrical model

• The cortex is a continuous-differentiable manifold • Fiber bundle • Lie symmetries of SE2 and sub-Riemannian structure • Neural activity is constrained by the structure

Amodal completion

in R2 ⇥ S1\⌃0

in R2 ⇥ S1\⌃0

Horizontal mean curvature flow

Horizontal Laplace-Beltrami flow

The amodal completion flow

Sarti, Citti 2003 Citti, Sarti 2006 !

ut =X11u(X2u)2 � 2X1uX2uX12u+X22u(X1u)2

(X1u)2 + (X2u)2

vt =X11v(X2u)2 � 2X1uX2uX12v +X22v(X1u)2

(X1u)2 + (X2u)2

Horizontal mean curvature flow

Horizontal Laplace-Beltrami flow

The amodal completion flow

vt =X11v(X2u)2 � 2X1uX2uX12v +X22v(X1u)2

(X1u)2 + (X2u)2 + ✏1+ ✏2�v

ut =X11u(X2u)2 � 2X1uX2uX12u+X22u(X1u)2

(X1u)2 + (X2u)2 + ✏1+ ✏2�u

See the poster for the proof of existence of the sub-Riemannian mean curvature flow of Citti-Sarti (2003,2006) and convergence of numerics.

Inpainting

Modal completion

Retinex model

Land et al 1974, Kimmel et al 2003, Morel et al. 2010

I(x, y) � log I

� log f = � log I

Retinex model

I(x, y)

h = log I

�h

�� = �h

� = log f

L1 =

Z|r��rh|2dxdy

The modal completion sub-Riemannian LagrangianZ

|r��rh|2dxdy +Z

|r�� ~

A|2dxdy +Z

|X1~

A|2dxdy

�� =1

2(�h+ div( ~A))

X11~A = �r�+ ~A

The Euler-Lagrange Equation

G.Citti, A.Sarti 2014

The field termX11

~A = �r�

The field term

X11~A = �r�

�� =1

2(�h+ div( ~A))

The particle term

�� =1

2(�h+ div( ~A))

The particle term

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Inverted contrast

37

Different apertures

Alternate polarity

Fragmentation

Koffka cross: narrow

Koffka cross: wide

The field term

X11~A = �r�

Constitution of perceptual units

Ermentraut-Cowan mean field equation of neural activity

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Horizontal connectivity kernel

� = X11!(⇠, 0) +X22!(⇠, 0)� = X1!(⇠, 0) +X22!(⇠, 0)

!(⇠, 0)

!(⇠, 0) ⇡ e�d2c(⇠,0)

The E-C equation in the domain of the input

The eigenvalue problem

sub-Riemannian kernel PCA

Z!(⇠, ⇠0)u(⇠0)d⇠0 = �̃ku

!(⇠i, ⇠j)ui = �̃kui

A.S., G.Citti, 2010,2014

Spectral decomposition: 1st eigenvector

Spectral decomposition: 2nd eigenvector

M.Favali, G.Citti, A.Sarti preprint 2014

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Seminar of Neuromathematics of Vision

European Institute of Theoretical Neuroscience!Paris

Organizers: G.Citti, A.Destexhe, O.Faugeras, J.P. Nadal, J.Petitot, A.Sarti